The Problem:
Given two fixed vectors $\mathbf{u}$ and $\mathbf{v}$, which conditions should the matrices $\mathbf{A}$ and $\mathbf{B}$ fullfill, for the following matrix equation to hold:
$\mathbf{u}^{H}\mathbf{B}\mathbf{v} = \mathbf{u}^{H}\mathbf{A}\mathbf{v}$
I suppose there is an infinite space of such matrices, but I would like to know whether this space has some special properties.
Note: assume that both vectors $u$ and $v$ are not zero, otherwise any $A$ and $B$ will do.
What you are asking is that the bilinear forms $A$ and $B$ take the same value on the couple $(u,v)$, but this only gives one constraint out of $n^2$ degrees of freedom.
Said differently, your condition says that after some congruence, $A$ and $B$ must have a common coefficient (one coefficient only, among $n^2$).
Hint: To find the change-of-coordinate matrix, complete $\{u, v\}$ into a basis of the vector space (you need to distinguish two cases according to whether $u$ and $v$ are colinear or not.
Edit: Let's write some details.
If $u$ and $v$ are collinear: Let $P$ be any invertible matrix whose first column is $u$. Consider $A' = P^T\, A \,P$ and $B' = P^T\, B \,P$. Then $u^T \, A \, v = u^T \, B \, v$ if and only if $A'_{11} = B'_{11}$.
If $u$ and $v$ are not collinear: Let $P$ be any invertible matrix whose first column is $u$ and second column is $v$. Consider $A' = P^T\, A \,P$ and $B' = P^T\, B \,P$. Then $u^T \, A \, v = u^T \, B \, v$ if and only if $A'_{12} = B'_{12}$.